       Pseudodecahedron

• To: mathgroup at smc.vnet.net
• Subject: [mg54010] Pseudodecahedron
• From: Roger Bagula <tftn at earthlink.net>
• Date: Mon, 7 Feb 2005 03:12:58 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```I found this surface last night.
It has the same angles and face number as a Dodecahedron
and is made by a parametric algorithm.
If you take the second Pi/5 rotation out, it is a
12 sided surface with ten quadralateral ( four sided) faces and two
pentagon faces. It uses a five sided and a 6 ( double 3) sided
function set. I call the result a psedodecahedron.
---------------------------------------------------------------
Mathematica notebook
(* 5 sided measure function*)
a5 = Max[Table[Cos[t + 2*Pi*n/5], {n, 0, 4}]]
(* six sided measure function*)
a3 = Max[Table[Abs[Cos[p + 2*Pi*n/3]], {n, 0, 2}]]
x = (Sin[p]/a3)*Sin[t]/a5;
y = (Sin[p]/a3)*Cos[t]/a5;
z = Cos[p]/a3;
(* rotated bottom section by Pi/5*)
x1 = x*Cos[Pi/5] - Sin[Pi/5]*y;
y1 = y*Cos[Pi/5] + Sin[Pi/5]*x;
z1 = z;
g = ParametricPlot3D[{x, y, z}, {t, -Pi, Pi}, {p, 0, Pi/2},
PlotPoints->60, Axes -> False, Boxed -> False];
g1 = ParametricPlot3D[{x1, y1, z}, {t, -Pi, Pi}, {p, Pi/2, Pi},
PlotPoints -> 60, Axes -> False, Boxed -> False];
Show[{g, g1}]
Show[{g, g1}, ViewPoint -> {-0.001, -0.416, -3.358}]
Show[{g, g1}, ViewPoint -> {-0.080, -3.375, -0.229}]
----------------------------------------------------------------
Respectfully, Roger L. Bagula